TeX macro for normal operator ordering

I've always been bothered with normal operator ordering, writing $:O(a)O(b):$ always produces bad results.

The quick fix I've been using is the following:

\def\normOrd#1{\mathop{:}\nolimits\!#1\!\mathop{:}\nolimits}

%%
% example:
% \begin{equation}
% \normOrd{a(z)b(\omega)} = a(z)_{+}b(\omega)+(-1)^{\alpha\beta}b(\omega)a(z)_{-}
% \end{equation}
%%

Which in practice looks like:

How I got this solution

I determined this solution iteratively after many different attempts, which I shall enumerate along with the problems they each had.

However, using mere colons :a(z)b(\omega): = ... produces the following:

Being clever, I asked myself "Hey, why not write :x\colon for the normal ordering?" This was clever, but wrong. Consider the following example:

g = :x\colon

Producing:

Not one to give up easily, I found a \cocolon definition on tex.stackexchange. Trying that instead:

g = \cocolon x\colon = y

Produces strange extra whitespace on the right:

After examining the co-colon code, I just determined that something along the lines of

% rough draft definition #1
\def\normOrd#1{\mathrel{:}\!#1\!\mathrel{:}}

would work. This didn't quite work, the whitespacing was strange. So instead I just use \mathop{:}\nolimits..., which produces the desired result.

Revising my Notes on General Relativity

So I've been revising my notes on general relativity, and I've found several things worth mentioning.

1. Equivalence Principle. The equivalence principle gives us geometry. This is often poorly described (I too committed this error in my drafts).

The equivalence principle tells us neither the composition of a body nor its mass determines its trajectory in a gravitational field. So gravity determines paths, and this gives us geometry.

Moreover, there are different equivalence principles which should be mentioned. I yielded to this, and became incoherent (alas!). The trick is to stick this into a box, for the interested reader to find out more about it, but not obstruct the writing.

2. Coordinates for Black Hole. Different coordinates for the Schwarzschild solution are described beautifully in Hans Stephani, Dietrich Kramer, Malcolm MacCallum, Cornelius Hoenselaers, and Eduard Herlt's Exact Solutions of Einstein’s Field Equations (Cambridge University Press, 2d edition, 2009).

3. Manifolds, Mathematics. I think I ought to examine Christopher Isham's Modern Differential Geometry for Physicists for a Physicist's differential geometry.

I should like to discuss the exponential map, which relates paths to geometry (as alluded in the equivalence principle discussion).

Most readers probably will agree that "Part II" of my notes (which specifically discuss differential geometry) are the toughest part of the notes.

Probably, I should mention a few examples of manifolds and explicitly study their coordinates in lecture 5.

3.1. Functions. I never discussed what it means for a function on a manifold (a) to exist, (b) to be smooth.

Really, this let us discuss curves too. Why? A curve is just a function γ:I→M where I is just a closed interval, and M is the manifold.

3.2. Diffeomorphisms. This word is thrown around a lot, but never defined rigorously (or at all!). So I should re-investigate this a bit.

Vertex algebras

Think about your favorite Lie algebra Lie(G). We have a mapping on it, namely, the adjoint representation:

ad:Lie(G) → End[Lie(G)]

where "End[Lie(G)]" are the endomorphisms of the Lie algebra Lie(G).

Normally this is of the form "ad(u)v∈Lie(G)" and is shorthand for "ad(u)=[u,-]".

The Jacobi identity looks like:

ad(u)ad(v)-ad(v)ad(u)=ad(ad(u)v).

This is the most important identity. Vertex operator algebras are an algebra with a similar property.

A vertex operator algebra consists of a vector space V equipped with a mapping usually denoted

Y:V→(End V)[[x,x^-1]].

In this form, it looks like left-multiplication operator...or that's the intuition anyways. So if "v∈V", we should think Y(v,x) belongs to "(End V)[[x,x^-1]]" and acts on the left.

Really through currying this should be thought of as "V⊗V→V[[x,x^-1]]", i.e., a sort of multiplication operator with a parameter "x". (This is related to the "state-operator correspondence" physicists speak of with conformal field theories.)

Just like a Lie algebra, the Vertex Operator algebra satisfies a Jacobi identity and it is the most important defining property for the VOA.

Lets stop and look at this structure again:

Y:V→(End V)[[x,x^-1]].

What's the codomain exactly? Well, it's a formal distribution (not a mere formal power series!).

So what does one look like? Consider δ(z-1) = Σ zⁿ where the summation ranges over n∈ℤ. This series representation is a formal distribution, and behaves in the obvious way. Lets prove this!

Desired Property: δ(z-1) vanishes almost everywhere.

Consider the geometric series f(z) = Σzⁿ where n is any non-negative integer (n=0,1,...).

Observe that δ(z-1) = f(z) + z^-1f(z^-1). Lets now substitute in the resulting geometric series:

δ(z-1) = [1/(1-z)] + z^-1[1/(1-z^-1)]

and after some simple arithmetic we see for z≠1 we have δ(z-1)=0.

Desired Property: for any Laurent polynomial f(z) we have δ(z-1)f(z)=δ(z-1)f(1).

This turns out to be true, thanks to the magic of infinite series; but due to html formatting, I omit the proof. The proof is left as an exercise to the reader (the basic sketch is consider δ(z-1)zⁿ, then prove linearity, and you're done).

Finite Field with Four Elements

Small note to myself on notational problems when facing finite groups.

Recall the finite field with four elements is ℤ₂[x]/(1+x+x²).

People often write ω = 1+x and ω=x. Observe then that ω² = ω, and ω² = ω. Moreover ωω=1 and 1+ω+ω=0.

I have only seen this ω notation specified in Pless' Error Correcting Codes, Third ed., page 102 et seq.

End notes and Foot notes in LaTeX

So I was writing up notes for a reading group on Afghanistan, and it has apparently become fashionable to use endnotes in the humanities. Being fond of Edward Gibbon, I use footnotes excessively. Irritating everyone, I use both while making it indistinguishable whether I refer to a footnote or endnote by a superscripted number.

How to do this in LaTeX? Quite simple:

\documentclass{article}

\usepackage{endnotes}

\makeatletter
\newcommand*{\dupcntr}[2]{%
    \expandafter\let\csname c@#1\expandafter\endcsname\csname c@#2\endcsname
}
\dupcntr{endnote}{footnote}
\renewcommand\theendnote{\thefootnote}
\makeatother

\begin{document}
Blah blah blah.
\end{document}

It turns out to work quite well.

I modified some macros from a TeX Stackexchange discussion on "slave counters"...so I get only partial credit.

Metapost and Labels

This is just a quick note to myself. When I want to write a label with a smaller font, I should use \scriptstyle...but it is tricky since it requires math mode!

So an example, consider this diagram describing an experiment for gravitational redshift:

numeric u;
u = 1pc;
beginfig(0)
  path earth;
  pair clock[];

  earth = fullcircle scaled u;

  clock[0] = (0,2u);
  clock[1] = (0,4u);

  draw (0,0)--(0,5u) dashed evenly;

  for i=0 upto 1:
    label(btex $\bullet$ etex, clock[i]);
  endfor;

  fill earth withcolor 0.75white;
  draw earth;
  label.rt(btex ${\scriptstyle\rm Earth}$ etex, (.5u,0));
  label.rt(btex ${\scriptstyle\rm Satellite\ 1}$ etex, clock[0]);
  label.rt(btex ${\scriptstyle\rm Satellite\ 2}$ etex, clock[1]);
endfig;
end;

Just remember to use "\ " for spaces. Otherwise it will all run together horribly!

Feynman Diagrams and Motives

I have been re-reading the following book:

Alain Connes and Matilde Marcolli,
Noncommutative Geometry, Quantum Fields, and Motives,
Colloquium Publications, Vol.55, American Mathematical Society, 2008.

It turns out that Dr Marcolli has taught a course on related material back in 2008! It is mostly dealing with the first chapter of the book.

Hopf Algebras and Feynman Calculations

There is a nice review of Hopf algebras used in Feynman diagram calculations:

Kurusch Ebrahimi-Fard, Dirk Kreimer,
"Hopf algebra approach to Feynman diagram calculations".
Eprint arXiv:hep-th/0510202v2, 30 pages.

For another specifically reviewing the noncommutative approach discussed in Connes and Matilde's book, see:

Herintsitohaina Ratsimbarison,
"Feynman diagrams, Hopf algebras and renormalization."
Eprint arXiv:math-ph/0512012v2, 12 pages.

What is a "Hopf algebra", anyways?

Pierre Cartier,
"A primer of Hopf algebras."
Eprint [math.osu.edu], 81 pages.

Hopf Algebras

What the deuce is a "Hopf algebra"? That's a very good question, and I'm very glad you asked it. Wikipedia has its definition, which may or may not be enlightening.

Lets consider a concrete example. Consider a finite group G, and the field of complex number ℂ. We assert the collection Hom(G,ℂ) is a Hopf algebra.

Recall we have multiplication of group elements. This is a mapping G×G→G.

Now, observe we have functoriality to give us a mapping Hom(G×G→G,ℂ) = ℂ^G→ℂ^G×ℂ^G. Lets call this thing Δ

Great, but what does it do? Good question!

Take some f∈Hom(G,ℂ) then what is Δ(f)?

It is a function of two variables, [Δ(f)](x,y). Functoriality demands, if we fix one of the arguments to be the identity element e∈G of the group, then [Δ(f)](e,y)=f(y) and [Δ(f)](x,e)=f(x).

It follows logically that [Δ(f)](x,y)=f(xy).

We also need to consider the antipode map S:ℂ^G→ℂ^G. We have [S(f)](x) be determined by the Hopf property, and a long story short [S(f)](x)=f(x^-1).

Note that the antipode map is a generalization of the "group inverse" notion.

The other algebraic structure is a triviality, lets consider other interesting applications!

Feynman Diagrams

Now, I have written some notes [pdf] on the basic algorithm evaluating Feynman diagrams and producing a number (the "probability amplitude").

As I understand it (and I don't!!) Ebrahimi-Fard and Kreimer suggest considering the Hopf algebra of "Feynman graphs" (which are just considered as colored graphs representing physical processes).

The basic algorithm to evaluating Feynman diagrams are based on the "Feynman rules" (what we assign to each edge, vertex, etc.). So Feynman rules are linear and multiplicative maps, associating to each Feynman graph (again, seen as a collection of vertices and edges) its corresponding Feynman integral.

So these maps are the important things, which enable us to algorithmically do stuff.

Lets stop! I said "Feynman integrals" are assigned to each graph...am I drunk, or is that correct?

Yes yes, the answer is "yes" ;)

What a horrible joke...but what I mean is: the scattering process of electrons, for example, is the infinite sum taking into account all the virtual processes.

Usually we only care up to a few orders.

Of course, this is my understanding of the Hopf algebra treatment of Feynman diagrams...and I openly admit: I could be completely wrong!

So to figure it out, I'll stop rambling, and continue reading.

(Ramblings on) Writing Notes on Quantum Field Theory

So, in the long run, my aim is to write great notes on quantum field theory and quantum gravity. Since quantum gravity depends on quantum field theory, it makes sense to begin there!

For those uninterested in my rambling thought process, here's the punch line: just as integrals and derivatives are first covered symbolically in calculus, then rigorously in analysis...we likewise believe that a naive and symbolic approach first ought to be covered, then a rigorous and axiomatic approach second.

Greiner and Reinhardt's Field Quantization provides a good level for detail, at least for the naive/symbolic treatment of field theory.

What Other People Do

The basic approach other books take is "Well, here's Feynman diagrams. Quantum Field Theory just plays with these...here's how you get Feynman rules...and here's renormalization, the end."

This is not terrible. But it is lacking a certain je ne sais quoi.

So instead, perhaps I should look at it from the mathematical perspective. This has its own problems.

Depends on...

The problem I have is with dependencies! It doesn't make sense to write about quantum field theory without first writing about classical field theory, quantum mechanics, and a bit about functional analysis.

I have written a note about Relativistic Quantum Mechanics [pdf] (which may make more sense after reading my notes on Lie Groups, Lie Algebras, and their Representations [pdf]).

However, there is still more to do with quantum mechanics. Particularly, the subject of scattering theory is lacking. (Erik Koelink has some great Lecture Notes [tudelft.nl] too)

Despite my notes on Functional Techniques in Path Integral Quantization [pdf], I still feel lacking in the "path integral" department.

Perhaps I should write notes on measure theory, functional analysis, then tackle Glimm and Jaffe's Quantum physics: a functional integral point of view?

With classical field theory, the subject quickly becomes a can of worms (sadly enough).

Gauge theory, as Derek Wise notes in his blog post "The geometric role of symmetry breaking in gravity", is intimately connected to Cartan Geometry.

There are dozens of exercises/examples to consider in gauge theory: Yang-Mills Theory, Born-Infield Action, Non-linear Sigma-Model, Non-linear Electrodynamics, Chern-Simons Theory, etc.

What Outline

So far, I've been considering my obstacles...but what about an outline?

The model I am following is the treatment of integration and differentiation in mathematics. First we have the naive symbolic manipulations (as done in calculus), then later we have the formal and rigorous proof based approach (as done in analysis).

Perhaps we should begin with naive field theory, where we obtain classical field theory "naively" from a "many body" problem.

This has merit from modelling fields as densities on the intuitive level.

Canonical quantization of this scheme becomes a triviality.

The problem with this approach is: what about the treatment of gauge theories, and their quantization?

After a few miracles, I expect to end up working with path integral quantization and formal calculus.

Naive treatment on quantizing gauge systems ought to be considered a bit more closely...

So that concludes the "naive" approach, and we begin the Axiomatic Approach. We should clarify the term "Axiom" means specification (not "God given truth", as dictionaries insist!).

The axiomatic approach would be done in a "guess-and-check" manner, modifying the axioms as necessary.

We naturally begin with Wightman axioms for the canonical approach, and the Osterwalder-Schrader axioms for the path integral approach. (Quickly, we ought to prove these two are equivalent!)

Kac's Vertex Algebras for Beginners takes the Wightman axioms, then extends them to conformal field theory through some magic. Perhaps this would be a worth-while example to consider?

Studying Tactics

My friend asked me for help on military tactics. "Where'd you get the idea I know anything about that?" I asked.

"I thought game theory was known to all mathematicians," my friend sheepishly replied.

Well, he's wrong about that. But I later found Clausewitz's suggestion on examples:

EXAMPLES from history make everything clear, and furnish the best description of proof in the empirical sciences. This applies with more force to the Art of War than to any other. General Scharnhorst, whose handbook is the best ever written on actual War, pronounces historical examples to be of the first importance, and makes an admirable use of them himself. Had he survived the War in which he fell, the fourth part of his revised treatise on artillery would have given a still greater proof of the observing and enlightened spirit in which he sifted matters of experience.

But such use of historical examples is rarely made by theoretical writers; the way in which they more commonly make use of them is rather calculated to leave the mind unsatisfied, as well as to offend the understanding. We therefore think it important to bring specially into view the use and abuse of historical examples.

[...]

Now, if we consider closely the use of historical proofs, four points of view readily present themselves for the purpose.

First, they may be used merely as an EXPLANATION of an idea. In every abstract consideration it is very easy to be misunderstood, or not to be intelligible at all: when an author is afraid of this, an exemplification from history serves to throw the light which is wanted on his idea, and to ensure his being intelligible to his reader.

Secondly, it may serve as an APPLICATION of an idea, because by means of an example there is an opportunity of showing the action of those minor circumstances which cannot all be comprehended and explained in any general expression of an idea; for in that consists, indeed, the difference between theory and experience. Both these cases belong to examples properly speaking, the two following belong to historical proofs.

Thirdly, a historical fact may be referred to particularly, in order to support what one has advanced. This is in all cases sufficient, if we have ONLY to prove the POSSIBILITY of a fact or effect.

Lastly, in the fourth place, from the circumstantial detail of a historical event, and by collecting together several of them, we may deduce some theory, which therefore has its true PROOF in this testimony itself.

For the first of these purposes all that is generally required is a cursory notice of the case, as it is only used partially. Historical correctness is a secondary consideration; a case invented might also serve the purpose as well, only historical ones are always to be preferred, because they bring the idea which they illustrate nearer to practical life.

The second use supposes a more circumstantial relation of events, but historical authenticity is again of secondary importance, and in respect to this point the same is to be said as in the first case.

For the third purpose the mere quotation of an undoubted fact is generally sufficient. If it is asserted that fortified positions may fulfil their object under certain conditions, it is only necessary to mention the position of Bunzelwitz [Frederick the Great's celebrated entrenched camp in 1761] in support of the assertion.

But if, through the narrative of a case in history, an abstract truth is to be demonstrated, then everything in the case bearing on the demonstration must be analysed in the most searching and complete manner; it must, to a certain extent, develop itself carefully before the eyes of the reader. The less effectually this is done the weaker will be the proof, and the more necessary it will be to supply the demonstrative proof which is wanting in the single case by a number of cases, because we have a right to suppose that the more minute details which we are unable to give neutralise each other in their effects in a certain number of cases.

If we want to show by example derived from experience that cavalry are better placed behind than in a line with infantry; that it is very hazardous without a decided preponderance of numbers to attempt an enveloping movement, with widely separated columns, either on a field of battle or in the theatre of war—that is, either tactically or strategically—then in the first of these cases it would not be sufficient to specify some lost battles in which the cavalry was on the flanks and some gained in which the cavalry was in rear of the infantry; and in the tatter of these cases it is not sufficient to refer to the battles of Rivoli and Wagram, to the attack of the Austrians on the theatre of war in Italy, in 1796, or of the French upon the German theatre of war in the same year. The way in which these orders of battle or plans of attack essentially contributed to disastrous issues in those particular cases must be shown by closely tracing out circumstances and occurrences. Then it will appear how far such forms or measures are to be condemned, a point which it is very necessary to show, for a total condemnation would be inconsistent with truth.

It has been already said that when a circumstantial detail of facts is impossible, the demonstrative power which is deficient may to a certain extent be supplied by the number of cases quoted; but this is a very dangerous method of getting out of the difficulty, and one which has been much abused. Instead of one well-explained example, three or four are just touched upon, and thus a show is made of strong evidence. But there are matters where a whole dozen of cases brought forward would prove nothing, if, for instance, they are facts of frequent occurrence, and therefore a dozen other cases with an opposite result might just as easily be brought forward. If any one will instance a dozen lost battles in which the side beaten attacked in separate converging columns, we can instance a dozen that have been gained in which the same order was adopted. It is evident that in this way no result is to be obtained.

Upon carefully considering these different points, it will be seen how easily examples may be misapplied.

An occurrence which, instead of being carefully analysed in all its parts, is superficially noticed, is like an object seen at a great distance, presenting the same appearance on each side, and in which the details of its parts cannot be distinguished. Such examples have, in reality, served to support the most contradictory opinions. To some Daun's campaigns are models of prudence and skill. To others, they are nothing but examples of timidity and want of resolution. Buonaparte's passage across the Noric Alps in 1797 may be made to appear the noblest resolution, but also as an act of sheer temerity. His strategic defeat in 1812 may be represented as the consequence either of an excess, or of a deficiency, of energy. All these opinions have been broached, and it is easy to see that they might very well arise, because each person takes a different view of the connection of events. At the same time these antagonistic opinions cannot be reconciled with each other, and therefore one of the two must be wrong.

Much as we are obliged to the worthy Feuquieres for the numerous examples introduced in his memoirs—partly because a number of historical incidents have thus been preserved which might otherwise have been lost, and partly because he was one of the first to bring theoretical, that is, abstract, ideas into connection with the practical in war, in so far that the cases brought forward may be regarded as intended to exemplify and confirm what is theoretically asserted—yet, in the opinion of an impartial reader, he will hardly be allowed to have attained the object he proposed to himself, that of proving theoretical principles by historical examples. For although he sometimes relates occurrences with great minuteness, still he falls short very often of showing that the deductions drawn necessarily proceed from the inner relations of these events.

Another evil which comes from the superficial notice of historical events, is that some readers are either wholly ignorant of the events, or cannot call them to remembrance sufficiently to be able to grasp the author's meaning, so that there is no alternative between either accepting blindly what is said, or remaining unconvinced.

It is extremely difficult to put together or unfold historical events before the eyes of a reader in such a way as is necessary, in order to be able to use them as proofs; for the writer very often wants the means, and can neither afford the time nor the requisite space; but we maintain that, when the object is to establish a new or doubtful opinion, one single example, thoroughly analysed, is far more instructive than ten which are superficially treated. The great mischief of these superficial representations is not that the writer puts his story forward as a proof when it has only a false title, but that he has not made himself properly acquainted with the subject, and that from this sort of slovenly, shallow treatment of history, a hundred false views and attempts at the construction of theories arise, which would never have made their appearance if the writer had looked upon it as his duty to deduce from the strict connection of events everything new which he brought to market, and sought to prove from history.

When we are convinced of these difficulties in the use of historical examples, and at the same time of the necessity (of making use of such examples), then we shall also come to the conclusion that the latest military history is naturally the best field from which to draw them, inasmuch as it alone is sufficiently authentic and detailed.

In ancient times, circumstances connected with War, as well as the method of carrying it on, were different; therefore its events are of less use to us either theoretically or practically; in addition to which, military history, like every other, naturally loses in the course of time a number of small traits and lineaments originally to be seen, loses in colour and life, like a worn-out or darkened picture; so that perhaps at last only the large masses and leading features remain, which thus acquire undue proportions.

If we look at the present state of warfare, we should say that the Wars since that of the Austrian succession are almost the only ones which, at least as far as armament, have still a considerable similarity to the present, and which, notwithstanding the many important changes which have taken place both great and small, are still capable of affording much instruction. It is quite otherwise with the War of the Spanish succession, as the use of fire-arms had not then so far advanced towards perfection, and cavalry still continued the most important arm. The farther we go back, the less useful becomes military history, as it gets so much the more meagre and barren of detail. The most useless of all is that of the old world.

But this uselessness is not altogether absolute, it relates only to those subjects which depend on a knowledge of minute details, or on those things in which the method of conducting war has changed. Although we know very little about the tactics in the battles between the Swiss and the Austrians, the Burgundians and French, still we find in them unmistakable evidence that they were the first in which the superiority of a good infantry over the best cavalry was, displayed. A general glance at the time of the Condottieri teaches us how the whole method of conducting War is dependent on the instrument used; for at no period have the forces used in War had so much the characteristics of a special instrument, and been a class so totally distinct from the rest of the national community. The memorable way in which the Romans in the second Punic War attacked the Carthaginan possessions in Spain and Africa, while Hannibal still maintained himself in Italy, is a most instructive subject to study, as the general relations of the States and Armies concerned in this indirect act of defence are sufficiently well known.

But the more things descend into particulars and deviate in character from the most general relations, the less we can look for examples and lessons of experience from very remote periods, for we have neither the means of judging properly of corresponding events, nor can we apply them to our completely different method of War.

Unfortunately, however, it has always been the fashion with historical writers to talk about ancient times. We shall not say how far vanity and charlatanism may have had a share in this, but in general we fail to discover any honest intention and earnest endeavour to instruct and convince, and we can therefore only look upon such quotations and references as embellishments to fill up gaps and hide defects.

It would be an immense service to teach the Art of War entirely by historical examples, as Feuquieres proposed to do; but it would be full work for the whole life of a man, if we reflect that he who undertakes it must first qualify himself for the task by a long personal experience in actual War.

Whoever, stirred by ambition, undertakes such a task, let him prepare himself for his pious undertaking as for a long pilgrimage; let him give up his time, spare no sacrifice, fear no temporal rank or power, and rise above all feelings of personal vanity, of false shame, in order, according to the French code, to speak THE TRUTH, THE WHOLE TRUTH, AND NOTHING BUT THE TRUTH.

Addendum: one may be interested in Napoleon's Maxims, Patton's Reading List, and Roosevelt's Pigskin Library.

Puzzles

Recently I've been more interested in puzzles.

Project Euler is the classic example of puzzles which require either higher math or computational skill (or both!).

Facebook has a collection of puzzles too, motivated from the engineering perspective.

But note: Facebook uses these puzzles for hiring people. Plus, the puzzles are not always mathematically oriented.

I suppose a good mathematician should always set up puzzles for themselves. As Socrates remarked:

SOCRATES: Indeed, Lysimachus, I should be very wrong in refusing to aid in the improvement of anybody. And if I had shown in this conversation that I had a knowledge which Nicias and Laches have not, then I admit that you would be right in inviting me to perform this duty; but as we are all in the same perplexity, why should one of us be preferred to another? I certainly think that no one should; and under these circumstances, let me offer you a piece of advice (and this need not go further than ourselves). I maintain, my friends, that every one of us should seek out the best teacher whom he can find, first for ourselves, who are greatly in need of one, and then for the youth, regardless of expense or anything. But I cannot advise that we remain as we are. And if any one laughs at us for going to school at our age, I would quote to them the authority of Homer, who says, that

'Modesty is not good for a needy man.'

Let us then, regardless of what may be said of us, make the education of the youths our own education. (Emphasis added, from Plato's Laches)

For example, I know a little bit about representations of Lie groups and Lie algebras (one can always learn more!)...but what about the representation of the quaternion group induced from the irreducible representations of SU(2)? How does it decompose into irreps? Etc.

Knuth remarked somewhere what helped him understand the representation theory for the symmetric group was writing a program which generated the permutation matrix representations.

I suspect writing a program which does these sorts of computations is a great puzzle for any mathematician that's savvy with programming.

Reading Material

And now, for something completely different.

A few papers I want to read:
When physics helps mathematics: calculation of the sophisticated multiple integral, 13 pages;
Some algebraic properties of differential operators, 15 pages;
Introduction to supergravity, 152 pages;
Fermionic impurities in Chern-Simons-matter theories, 31 pages;
Spinors and Twistors in Loop Gravity and Spin Foams, 16 pages;
Quaternionic Analysis, Representation Theory and Physics, 60 pages.

Problem Notebooks

I've been looking for the infamous Kourovka notebook, but there are apparently others.

Recall the Kourovka notebook is a collection of open problems in group theory. There are other notebooks with open problems in other fields.

For example, the Dniester Notebook [usask.ca] (pdf, 56 pages) has problems in ring theory.

The notebook stopped being published years ago. Now it's been open sourced.

The Sverdlovsk notebook stopped publishing back in 1989; it was a collection of open problems for semigroups, but it cannot be found.

Other problem notebooks might exist too, but I am unaware of them...

People don't read anymore :(

I have been wandering around book stores recently, and stumbled upon the Landmark Herodotus. If you recall, I wrote a few posts on Herodotus.

People ought to be reading the text, and produce their own notes, producing a similar result (as Landmark) which is personalized.

Don't mistake me: I think Landmark is a wonderful resource! It has helped with maps, and so on...but people should be doing this on their own.

The problem Landmark posed (to me) parallels Cliff Notes. The books are produced as an example of how to take notes while reading books...not a replacement for reading!

Extemporaneously producing maps is an invaluable skill. The map doesn't have to be precise, e.g. drawing Asia minor as a rectangle, or Greece as three rectangles.

There are some elements to, e.g., the Landmark Herodotus that are pleasant, and makes it a great resource to borrow from. For example: what did Herodotus get wrong?

It appears that he got a lot of Egyptian history wrong, for example.

But my point is: reading is studying! You have to make a book your own through notes, maps, references to other books, etc.

How you study is entirely up to you; how you take notes, well, that is entirely up to you. But reading without taking notes is like dancing: nobody does it unless drunk or insane.

People just confuse the mechanics of reading (parsing letters into words) with the procedure called "reading" (obtaining information from books, and evaluating it). *sigh*

P.S. Happy New Year!